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...Cse probabilistic analysis and randomized algorithms lecturer hung q ngo sunyatbualo spring last update april discrete time markov chains examples chain dtmc is an extremely pervasive probability model in this lecture we shall briey overview the basic theoretical foundation of let us rst look at a few which can be naturally modelled by example gambler ruin problem has he bets each game wins with stops playing gets broke or natural questions include what s that on average how many games are played special case so called using as follows will hand wavy before rigorously dening imagine have states shown figure state i indexed number dollars having equal move up to down for single server queue slot internet router buer addi tional packet p releases one remains same r starting from empty distribution packets after n slots overowed typical size these types answered denitions properties astochastic process collection random variables some space set t x when think points denoted all possible v...